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Introduction

Customers around the world flock to Zulily on a daily basis to discover and shop products that are curated specially for them. To serve our customers better, we are constantly innovating to improve the customer experience. To help us decide between different customer experiences, we primarily use split testing or A/B testing, the industry standard for making scientific decisions about products. In a previous post, we talked about the fundamentals of A/B testing and described our home-grown solution. In this article, we are going to explore one choice behind designing sound A/B tests: we will calculate the number of people that need to be included in your test.

Figure 1. For example, in an A/B test we might compare two variants of the navigation bar on the website. Here, the New Today tab, displaying new events, appears either in position one (original version) or position two (test version).

To briefly recap the idea of A/B testing, in an A/B test we compare the performance of two variants of the same experience. For example, in a test we might compare two ways of ordering the navigation tabs on the website (see Figure 1). In the original version (variant A or control), the tabs are ordered such that events launched today (New Today) appear before events that are ending soon (Ends Soon). In the new version of the website (variant B or control), Ends Soon appears before New Today. The test is set up such that, for a pre-defined period of time, customers visiting the website would be shown either variant A or variant B. Then, using statistical methods, we would measure the incremental improvement in the experience of customers that were shown variant B over those who were shown variant A. Finally, if there was a statistically significant improvement, we might decide to change order of the tabs on the website.

Since Zulily relies heavily on A/B
testing to make business decisions, we are careful about avoiding common
pitfalls of the method. The length of an A/B test ties in strongly with the
success of the business for two reasons:

If A/B tests run for more days than necessary,
the pace of innovation at the company will slow down.

If A/B tests run for fewer days than required
to achieve statistically sound results, the results will be misguided.

To account for these issues, before making decisions based on the A/B test results, we run what’s called a ‘power analysis.’A power analysis ensuresthat a certain number of people have been captured in the test to confirm or deny whether variant B was an improvement over variant A, which is the focus of this article. We also make sure that the test is run long enough so that short-term business cycles are accounted for. The calculation for the number of people needed in a test is a function of three things, effect size, significance level (), and power ().

To consult the statistician after an experiment is finished is often merely to ask [them] to conduct a post mortem examination. [They] can perhaps say what the experiment died of.

– Robert Fisher, Statistician

Common Terms

Before we get into the mechanics
of that calculation, let us familiarize ourselves with some common statistical
terms. In an A/B test, we are trying to estimate how all the customers behave (population)
by measuring the behavior of a subset of customers (sample). For
this, we ensure that our sample is representative of our entire customer base. During
the test, we measure the behavior of the customers in our sample. Measurements
might include the number of items purchased, the time spent on the website, or
the money spent on the website by each customer.

For example, to test whether
variant B outperformed variant A, we might want to know if the customers
exposed to variant B spent more money than the customers exposed to variant A.
In this test, our default position is that variant B made no difference on the behavior
of the customers when compared to variant A (null hypothesis). As
more customers are exposed to these variants and start purchasing products, we
collect measurements on more customers, which allows us to either accept or
reject this null hypothesis. The difference between the behavior of customers
exposed to variant A and variant B is known as the effect size.

Figure 2. In A/B testing, different types of errors can occur, depending on where the results lie on this graph. Therefore, the parameters that we set under the hood, namely significance level and power, need to be set carefully, keeping our appetite for error in mind.

Further, there are a number of parameters set under the hood. Before starting the test, we assign a significance level () which means that we might reject the null hypothesis when it is actually true in 5% of the cases (Type I error rate). Further, we will assign a power () which means that when the null hypothesis does not hold, or variant B changes the behavior of customers, the test will allow us to reject the null hypothesis 80% of the time. Importantly, these parameters need to be set at the beginning of the test and upheld for the duration of the test to avoid p-hacking, which leads to misguided results.

Figure 2. In A/B testing, different types of errors can occur, depending on where the results lie on this graph. Therefore, the parameters that we set under the hood, namely significance level and power, need to be set carefully, keeping our appetite for error in mind.

Estimating the Number of Customers for the A/B Test

For this exercise, let us revisit the previous example where we show customers two versions of the Zulily navigation bar. Let us say we want to see if this change makes customers less or more engaged on Zulily’s website. One metric that can capture this is the proportion of customers who revisit the website when shown variant B versus variant A. for this exercise, let us say that we are interested in a boost in this metric of at least 1 % (effect size). If we see at least this effect size, we might implement the variant B. Question is, how many customers should be exposed to each variant to allow us to confirm that this change of 1% exists?

Starting off, we define some parameters. First, we define the significance level at 0.05. Second, from the central limit theorem, we assume that the average money that a group of customers spend on the website is normally distributed. Third, we direct 50% the customers visiting the site to variant A and 50% to variant B. These last two points greatly simplify the math behind the calculation. Now, we can estimate the number of people that need to be exposed to each variant.

where, is the standard deviation of the population, is the change we expect to see, the , are quantile values calculated from a normal distribution. For the case of the parameters defined above, significance level of 0.05 and power of 0.80, and if we wanted to detect a 1% change in the proportion of people revisiting the website, our formula would simplify to:

This formula gives us the number of
people that need to be exposed to one variant. Finally, since the customers
were split evenly between variant A and variant B, we would need twice the
number of the people in the entire test. This estimate can change significantly
if any of the parameters change. For example, to detect a larger difference at
this significance level, we would need much smaller samples. Further, if the
observations are not normally distributed, then we would need a more
complicated approach.

Benefits of this calculation

In short, getting as estimate of
the number of customers needed allows us to design our experiments better. We
suggest conducting a power analysis both before and after starting a test for
several reasons:

Before starting the test – This gives us
an estimate of how long our test should be run to detect the effect that we are
anticipating. Ideally, this is done once to design the experiment and the
results are tallied when the requisite number of people are exposed to both
variants. However, the mean and standard deviation used in the calculation
before starting the test are approximations to the actual values that we might
see during the test. Thus, these a priori estimates might be off.

After starting the test – As the test
progresses, the mean and standard deviation converge to values representative
of the current sample which allows us to get more accurate estimates of the
sample size. This is especially useful in cases where the new experience
introduces unexpected changes in the behavior of the customers leading to significantly
different mean and standard deviation values than those estimated earlier.

Conclusion

At Zulily, we strive to make well-informed choices for our customers by listening to their voice through our A/B testing platform, among other channels, and ensuring that we are constantly serving the needs of the customers. While obtaining an accurate estimate of the number of people for the test is challenging, we hold it central to the process. Most people agree that the benefits of a well-designed, statistically sound A/B testing system far outweigh the benefits from obtaining quick, but misdirected numbers. Therefore, we aim for a high-level level of scientific rigor in our tests.

I would like to thank my colleagues in the data science team, Demitri Plessas and Pamela Moriarty, and my manager, Paul Sheets, for taking time to review this article. This article is possible due to the excellent work by the entire data science team of maintaining the A/B testing platform, and ensuring that experiments at Zulily are well-designed.